Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.

f1: Discretization of the FrFT.(a) Pictorial view of actual fractional Fourier transforms exemplified as continuous rotations in phase space. (b) Schematic representation of a pre-engineered Jx-array. (c–e) Top views of continuous ‘rotations' of a rectangular (c), displaced rectangular (d) and Gaussian (e) optical wave functions in a Jx-array with N=151. The bottom and top plots show the intensities of the ingoing and outgoing wave packets, respectively. The green lines describe the magnitude of phase distributions of the optical fields, that is, the phase jumps of π due to a change of sign of the signal's amplitude are not shown.

Mentions:
Similarly to its continuous counterpart, the DFrFT can be interpreted physically as a continuous rotation of the associated wave functions through an angle Z in phase space (see Fig. 1a)18. The idea is thus to construct finite circuits that are capable of imprinting such rotations to any light field. In quantum mechanics, three-dimensional spatial rotations of complex state vectors are generated via operations of the angular momentum operators Jk (k=x, y, z) on the Hilbert space of the associated system19. In particular, the rotation imprinted by the Jx-operator turns out to be an elaborated definition of the DFrFT (see Methods section for discussion). These concepts can be readily translated to the optical domain by mapping the matrix elements of the Jx-operator over the inter-channel couplings of engineered waveguide arrays (Fig. 1b–e)20. The coupling matrix of such waveguide arrays is thus given by (ref. 19). Here, κ0 is a scaling factor introduced for experimental reasons. The indices m and n range from −j to j in unit steps. Meanwhile, j represents an arbitrary positive integer or half-integer that determines the total number of waveguides via N=2j+1 (Fig. 1b).

f1: Discretization of the FrFT.(a) Pictorial view of actual fractional Fourier transforms exemplified as continuous rotations in phase space. (b) Schematic representation of a pre-engineered Jx-array. (c–e) Top views of continuous ‘rotations' of a rectangular (c), displaced rectangular (d) and Gaussian (e) optical wave functions in a Jx-array with N=151. The bottom and top plots show the intensities of the ingoing and outgoing wave packets, respectively. The green lines describe the magnitude of phase distributions of the optical fields, that is, the phase jumps of π due to a change of sign of the signal's amplitude are not shown.

Mentions:
Similarly to its continuous counterpart, the DFrFT can be interpreted physically as a continuous rotation of the associated wave functions through an angle Z in phase space (see Fig. 1a)18. The idea is thus to construct finite circuits that are capable of imprinting such rotations to any light field. In quantum mechanics, three-dimensional spatial rotations of complex state vectors are generated via operations of the angular momentum operators Jk (k=x, y, z) on the Hilbert space of the associated system19. In particular, the rotation imprinted by the Jx-operator turns out to be an elaborated definition of the DFrFT (see Methods section for discussion). These concepts can be readily translated to the optical domain by mapping the matrix elements of the Jx-operator over the inter-channel couplings of engineered waveguide arrays (Fig. 1b–e)20. The coupling matrix of such waveguide arrays is thus given by (ref. 19). Here, κ0 is a scaling factor introduced for experimental reasons. The indices m and n range from −j to j in unit steps. Meanwhile, j represents an arbitrary positive integer or half-integer that determines the total number of waveguides via N=2j+1 (Fig. 1b).

Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.